Is it possible to have overnight stay at Airport? If yes, then which terminal

Hotel near by

Hotel and free shuttle probably safer options.

There's a Holiday Inn by the airport. Free shuttle. I have seen people staying overnight at the airport, any terminal. But, it's not comfortable.

yes

I don't know if there is a hotel at the airport but Newark international airport Marriott hotel offers free shuttle service from any terminal 24 hours 7 days.